As of spring 2020, this site is dynamically generated from researchseminars.org, which see for other seminars worldwide (or for this seminar listed in your local timezone).
|
March 31, online |
Organizational meeting (Zoom only) |
|
April 7 |
David Hansen (MPIM)
In joint work with Lucas Mann, we establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we prove a duality theorem, establish bounds on Gelfand-Kirillov dimension, prove some non-vanishing results, and show a kind of partial Künneth formula. The key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by Mann. |
|
April 14, 10:00AM, online |
Amit Ophir (Hebrew U.)
Motivated by questions about a p-adic Fourier transform, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrodinger representations are explicit irreducible smooth representations that play an important role in their representation theory. Classically, the field of coefficients is taken to be the complex numbers and, among other things, one studies the unitary completions of the representations (which are well understood). By taking the field of coefficients to be an extension of the p-adic numbers, we can consider completions that better capture the p-adic topology, but at the cost of losing the Haar measure and the $L^2$-norm. Nevertheless, we establish a rigidity property for a family of norms (parametrized by a Grassmannian) that are invariant under the action of the Heisenberg group. The irreducibility of some Banach representations follows as a result. The proof uses "q-arithmetics". |
|
April 21 |
Anthony Kling (U. Arizona)
Let $N\geq3$ and $r\geq1$ be integers and $p\geq2$ be a prime such that $p\nmid N$. One can consider two different integral structures on the space of modular forms over $\mathbb{Q}$, one coming from arithmetic via $q$-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply results due to Brian Conrad to the situation of modular forms of even weight and level $\Gamma(Np^{r})$ over $\mathbb{Q}_{p}(\zeta_{Np^{r}})$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^{r}$ whenever $p^{r}>3$, allowing us to compute a lower bound which agrees with the upper bound. Hence we are able to compute the exponent precisely. |
|
April 28 |
Brian Lawrence (UCLA)
Interesting moduli spaces don't have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than H^{\epsilon}, for any positive \epsilon. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X. Joint with Ellenberg and Venkatesh. |
|
May 5 |
Masahiro Nakahara (U. Washington)
A variety satisfies potential density if it contains a dense subset of rational points after extending its ground field by a finite degree. A collection of varieties satisfies uniform potential density if that degree can be uniformly bounded. I will discuss this property for connected algebraic groups of a fixed dimension and elliptic K3 surfaces. This is joint work with Kuan-Wen Lai. |
|
May 12 |
Gyujin Oh (Princeton)
We interpret a harmonic Maass form as a variant of a local cohomology class of the modular curve. This is not only amenable to algebraic interpretation, but also nicely generalized to other Shimura varieties, avoiding the barrier of Koecher's principle, which could be useful for developing a generalization of Borcherds lifts. In this talk, we will exhibit how the theory looks like in the case of Hilbert modular varities. |
|
May 19 |
Michelle Manes (U. Hawaii) (paper)
In classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb R$ or $\mathbb C$ (or more generally self-maps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of self-maps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to post-critically finite rational maps. This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about pre-images of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational pre-images and open image results for Galois representations associated to dynamical systems. |
|
May 26 |
Koji Shimizu (UC Berkeley)
We will discuss a $p$-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic $p$. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties. |
|
June 2 |
Alexandra Florea (UC Irvine)
I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui. |